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Home , Algebraic group, Linear algebraic group

DENSE ORBITS AND DOUBLE COSETS

JONATHAN BRUNDAN Department of Pure 16, Mill Lane Cambridge, CB2 1SB

Introduction Throughout, G denotes a connected reductive algebraic defined over an algebraically closed field k of p 0. We are mainly inter- ested here in positive characteristic. ≥ In his article in these proceedings, Seitz considers the problem of finding closed H and P of G for which there are finitely many H,P - double cosets in G; of particular interest is the case when H is reductive and P is parabolic. We will describe some results on the closely related (and often easier) problem of finding such subgroups H and P with a dense H,P -double coset in G. We mention two special cases: first, let ρ : H G = GL(V ) be a rational representation of H and let P be the stabilizer→ in G of a 1-space in V . Then, there is a dense ρH, P -double coset in G if and only if H has a dense orbit on the set of 1-subspaces of V – that is, V is a prehomogeneous space for H. Irreducible prehomogeneous spaces have been classified in [26] (p = 0) and [5], [6] (p > 0). Secondly, take G to be arbitrary but let P = B be a Borel of G. Then, subgroups H of G with a dense orbit on the flag variety G/B – equivalently, a dense H,B-double coset – are called spherical subgroups, and the associated varieties G/H are spherical varieties (see [3] for a sur- vey of some geometric results). In characteristic 0, all reductive spherical subgroups of G have been classified, by Kr¨amer[16] (G simple) and Brion [2], Mikityuk [20] (G reductive); there is at present no such classification in positive characteristic. One aim of this article is to make precise a reduction modulo p argu- ment, assuming that the subgroup H is ‘defined over ’ in a suitable way. This argument depends on a representation-theoretic criterion for the 260 existence of a dense H,P -double coset due to Kimel’fel’d and Vinberg, which we review in section 1. The reduction modulo p argument described in section 2 will sometimes allow us to deduce the existence of a dense H,P -double coset in character- istic p from (often known) existence in characteristic 0. For this to work, we need the subgroup H to satisfy an extra condition connected with good filtrations; we give some examples (and non-examples) of subgroups satisfy- ing this extra condition in section 3. In particular, by a result of Donkin and Mathieu, all Levi subgroups satisfy the condition, so we obtain the classifi- cation of all spherical Levi subgroups immediately from Kr¨amer’sclassifi- cation over C. The problem of classifying more general spherical subgroups is discussed further in section 4. We conclude the introduction with some remarks on the relationship between the existence of a dense H,P -double coset as discussed here and the existence of finitely many double cosets as in Seitz’s problem. Of course, if there are finitely many H,P -double cosets in G then there is a dense one, but the converse is false in general. For instance, take H to be the radical of the opposite parabolic subgroup to P : there is always a dense H,P -double coset in G, but there are finitely many double cosets if and only if the of P is a union of connected components of the Dynkin diagram of G. However, in the special case that P = B is a of G, the existence of a dense H,B-double coset in G does imply that there are finitely many H,B-double cosets in G. This follows from the following remarkable result (with X = G/H), due to Brion [1] and Vinberg [29] in characteristic 0 and recently extended to positive characteristic by Knop [15, Corollary 2.6]: Finiteness Theorem Suppose that X is an irreducible G-variety on which B has a dense orbit. Then, B has finitely many orbits on X. So, the reductive spherical subgroups give a family of examples in Seitz’s finite orbit problem.

1. Representation-theoretic interpretation In this section, we review results of Kimel’fel’d and Vinberg [14] giving representation-theoretic criteria for the existence of dense orbits. These results were originally proved only in characteristic 0, but generalize quite easily to arbitrary characteristic, as is well known to several authors. First, we set up notation that will be in place for the remainder of the article. Always, G will denote a connected reductive with fixed Borel subgroup B containing a maximal torus T . Let X(T ) denote the of T , W = NG(T )/T denote the , and choose 261

R a W -invariant inner product , on Z X(T ). Let Φ X(T ) denote the h i ⊗ 2α ⊂ of G. For α Φ, α denotes . Let αi i I be a base ∈ ∨ α,α { | ∈ } for Φ, chosen so that B contains all negativeh iroot subgroups. For α Φ, ∈ let Uα denote the corresponding T -root subgroup of G and for a subset J of I, let PJ denote the standard parabolic subgroup B,Uαj j J . Let X(T )+ X(T ) denote the dominant weights,h i.e. the| ∈ weightsi λ ⊂ satisfying λ, α 0 for all i I. Let ωi i I denote the fundamental h i∨i ≥ ∈ { | ∈ } dominant weights, satisfying ωi, αj∨ = δij for all i, j I. Our labelling of h i + ∈ Dynkin diagrams is as in [11]. For λ X(T ) let G(λ) denote the dual G ∈ ∇ Weyl module indBλ, and let ∆G(λ) denote the corresponding Weyl module, which is the contravariant dual of G(λ). Given an algebraic group S and∇ an irreducible S-variety X, we let k(X) 1 denote the function field of X; S acts on k(X) by (g.f)(x) = f(g− x) for f k(X) defined at x X, and g S. We let k(X)S denote the S-fixed points∈ in k(X). We have∈ the basic invariant-theoretic∈ fact: Lemma 1.1 (Rosenlicht, [25]) The transcendence degree of k(X)S over k equals the minimum codimension of an S-orbit in X. In particular, S has a dense orbit on X if and only if k(X)S = k. Now we give the two results of Kimel’fel’d and Vinberg. The first is [14, Theorem 2]; we have included the proof since it is quite instructive. Theorem 1.2 Let X be an irreducible affine G-variety. Then, B has a dense orbit on X if and only if HomG(∆G(λ), k[X]) is at most 1-dimensional for all λ X(T )+. ∈ Proof. For notational convenience, we work with the opposite Borel sub- + group B to B. Suppose that HomG(∆G(λ), k[X]) is at least 2-dimensional for some λ. Then, we can find linearly independent functions f, g k[X] that are B+-high weight vectors of weight λ. But then the quotient∈f/g k(X) is a non-constant B+-invariant, so there is no dense B+-orbit on X∈. Conversely, suppose that there is no dense orbit, so that we can find a non-constant B+-invariant function φ k(X)B+ by Lemma 1.1. As X is affine, we can write φ = f/g for f, g ∈k[X]. The B+-submodule V of ∈ + k[X] generated by f is finite-dimensional, so we can find b1, . . . , bn B +∈ such that V = b1.f, . . . , bn.f . Now, by the Lie-Kolchin theorem, B fixes h i a 1-space in V , so there are scalars ci k such that f˜ := cibi.f spans a + ∈ P B -stable 1-space, of weight λ say. Letg ˜ = cibi.g. Since f/g is B+-invariant, f(b.g) = g(b.f)P for all b B+, so gf˜ = fg˜, so f/˜ g˜ = f/g. This is B+-invariant, so as b.f˜ = λ(b)f˜ for∈ all b B+, the same is true forg ˜. But φ is non-constant, so f˜ andg ˜ are linearly independent∈ B+- high weight vectors in k[X] of high weight λ. Now the universal property of Weyl modules implies that dim HomG(∆G(λ), k[X]) 2. ≥ 262

Recall that a subgroup H of G is spherical if it has a dense orbit on G/B. As a corollary to the theorem, we obtain a representation-theoretic criterion for reductive spherical subgroups: Corollary 1.3 Let H be a closed, connected reductive subgroup of G. Then, H is spherical if and only if the fixed point space

H G(λ) ∇ is at most 1-dimensional for all λ X(T )+. ∈ Proof. There is a dense H-orbit on G/B if and only if there is a dense B-orbit on G/H. Now we note that the variety G/H is an affine variety as H is reductive, by [22], and apply Theorem 1.2 to deduce that there is a dense B-orbit on G/H if and only if HomG(∆G(λ), k[G/H]) is at most 1-dimensional for all λ X(T )+. To complete the proof, the G-module ∈ G k[G/H] is precisely the induced module indH k, so by Frobenius reciprocity [13, I.3.4], HomG(∆G(λ), k[G/H]) ∼= HomH (∆G(λ), k). The result follows on dualizing. Kimel’fel’d and Vinberg also prove a second result [14, Theorem 1], which gives a similar criterion for the existence of a dense H,P -double coset in G, for any closed subgroup H and any parabolic P . A proof in arbitrary characteristic can be found in [4, Theorem 3.5]; the idea is sim- ilar to the proof of Theorem 1.2, this time exploiting the fact that if G is a semisimple, simply-connected algebraic group, then k[G] is a unique factorization domain [21].

Theorem 1.4 Let H be a closed, connected subgroup of G. Let P = PJ be a parabolic subgroup of G. There is a dense H-orbit on G/P if and only if the fixed point space H ( G(λ) µ) ∇ ⊗ is at most 1-dimensional for all 1-dimensional H-modules µ and all λ X(T )+ such that λ, α = 0 for all j J. ∈ h j∨i ∈ Corollary 1.3 and Theorem 1.4 are particularly useful in proving the non-existence of a dense double coset. For example, it is known that the Levi subgroup H = GLn(k)GLm(k) < GLn+m(k) = G is a spherical subgroup. In fact, providing n = m, the derived subgroup H = SLn(k)SLm(k) is 6 0 also spherical, but this is false if n = m since then H0 fixes a 2-dimensional subspace in the dual Weyl module n V , where V is the natural G-module. On the other hand, the existenceV of dense double cosets can often be proved directly. I am grateful to R. Guralnick for allowing me to include his arguments in the following examples which we will refer to later: 263

Lemma 1.5 (Guralnick) The following are spherical in all characteristics: (i) SOn(k) < SLn(k) for n 2; ≥ (ii) SOn(k)SOm(k) < SOn+m(k) for n, m 1. ≥ Proof. Let V be a vector space endowed with a q. We say that W V is a non-singular subspace if either the restriction to W of the bilinear form≤ coming from q is non-degenerate, or W is odd-dimensional in characteristic 2 and the bilinear form on W has a 1-dimensional radical R on which q is non-zero. (i) Let F be a maximal flag for SLn(k) so that each subspace is non- singular. Let B be the Borel subgroup stabilizing this flag. We claim that SOn(k) B is finite, which suffices to show that SOn(k) is spherical by ∩ . Consider the 2-dimensional subspace V2 in the flag; B induces a subgroup of O2 on this subspace, acting as 1 on a non-singular vector. ± This is a finite subgroup of O2. Now, B SOn(k) also preserves V2⊥, so by ∩ induction B SOn(k) is finite as desired. (ii) Use induction∩ on n. If either n = 1 or m = 1, we are done by [4, Theorem A]. So suppose n, m 2. Let V be the natural Weyl module for ≥ G = SOn+m(k), with quadratic form q. Suppose H = SOn(k)SOm(k) is the stabilizer in V of a non-singular n-dimensional subspace W < V . Pick a totally singular line R such that R⊥ W is a non-singular subspace of W of dimension n 1. Let P be the parabolic∩ subgroup of G fixing R. Now, S = P H stabilizes− R W, R and W , so Witt’s theorem easily implies ∩ ⊥ ∩ that S ∼= SOn 1(k)SOm 1(k). Given this, by we see that HP is dense in G. Now− let L be− a Levi subgroup of P containing S. By induction, there is a Borel subgroup BS < S such that SBS is dense in L. So if V is the unipotent radical of P , B = BSV is a Borel subgroup of G and SB is dense in P . So, HB = HSB is dense in HP , which is dense in G.

2. Integral embeddings We now consider reduction modulo p. We begin by recalling some basic facts about group schemes; we refer the reader to [13, I.2, II.1] for more details of these matters. Let R be a commutative and S be an R-group (as in [13, I.2.1]); so S is the associated to a commutative over R, namely its coordinate ring R[S]. We say S is an algebraic R-group if R[S] is finitely presented, reduced if R[S] is reduced, and flat if R[S] is. Given an R-algebra K and an R-module M, MK denotes M K; if M is ⊗R an S-module (equivalently, an R[S]-comodule) then MK is an SK -module. If K is algebraically closed and SK is a reduced algebraic K-group, we may regard it simply as a over K, written SK . We recall the universal coefficient theorem: 264

Proposition 2.1 ([13, I.4.18]) Suppose that R is a Dedekind ring and that S is a flat R-group scheme. Let K be an R-algebra. Then, for any torsion free S-module M and any i N, there is an exact sequence: ∈ i i R i+1 0 H (S,M)K H (SK ,MK ) Tor (H (S,M),K) 0. −→ −→ −→ 1 −→ Given our fixed connected reductive algebraic group G over k, there is Z a split, reductive algebraic -group G such that Gk ∼= G. For any ring R, we obtain the scheme GR on base change; note that R[GR] is R-free, so GR is flat. Fix now a pair (G, H) of connected reductive algebraic groups over k, where H is a closed subgroup of G, and let G,H denote corresponding split schemes over Z. We say that (G, H) is an integral embedding (defined over R) if there is an embedding i : HR GR and an isomorphism θ : Gk G → → such that θ ik : Hk G is an isomorphism between Hk and H, for some Dedekind ring◦ R with→ field of fractions of characteristic 0 for which k is an R-algebra. Given any R-algebra K that is an algebraically closed field, we let (GK ,HK ) be the corresponding embedding over K, and we identify (Gk,Hk) with (G, H). The following give examples of integral embeddings: (1) any semisimple subgroup H of G = SL(V ), where H and the H- module V arise via the Chevalley construction; (2) any subgroup corresponding to a closed subsystem of the root system of G; (3) centralizers of graph of G; (4) SOn(k)SOm(k) < SOn+m(k). In fact, all these cases are integral embeddings defined over Z, arising from embeddings of the Kostant Z-forms U(h)Z U(g)Z of the correspond- ing semisimple Lie algebras over C. For more general→ integral embeddings, R may need to be a ring of integers in an algebraic number field. Now we obtain the following statement on reduction modulo p: Theorem 2.2 Suppose that H is a closed, connected reductive subgroup of G such that (G, H) is an integral embedding defined over R. Let K be the of the field of fractions of R (of characteristic 0). Let P be a standard parabolic subgroup of G and let PK be the corresponding parabolic subgroup of GK . (i) If there is no dense HK ,PK -double coset in GK , there is no dense H,P -double coset in G. 1 (ii) If there is a dense HK ,PK -double coset in GK and H (H, G(λ) µ) = 0 for all λ, µ as in Theorem 1.4, then there is a dense H,P∇-double⊗ coset in G. 1 (iii) If HK is a spherical subgroup of GK and H (H, G(λ)) = 0 for all λ X(T )+, then H is spherical in G. ∇ ∈ 265

Proof. By localizing if necessary, we may assume that R is a DVR with irreducible element π so that k is an extension of the residue field R/πR. Let HR < GR denote the corresponding schemes over R. We first claim: H H For an R-free HR-module M of finite rank, dimk(Mk) k dimK (MK ) K . 1 ≥ If H (Hk,Mk) = 0 then equality holds. To prove this, we may assume that k actually equals the residue field H H R/πR. Note that R is a PID so M R is R-free, of rank d = dimK (MK ) K as K is flat over R. Proposition 2.1 with i = 0 implies that M HR k H H ⊗ embeds injectively into (Mk) k , so that dimk(Mk) k d. For the second 1 1 ≥ part, let A = H (HR,M) and suppose that H (Hk,Mk) = 0. We just need R HR Hk to show that Tor1 (A, k) = 0, since then M k ∼= (Mk) . We know by our assumption and Proposition 2.1 with i =⊗ 1 that A k = 0. But ⊗ A k = 0 implies A R/πR ∼= A/πA = 0, so π : A A is surjective. Since⊗ A is a finitely generated⊗ R-module (which follows→ from [8, 1.7]), the Nakayama lemma now implies that π : A A is an isomorphism. Finally, R → Tor1 (A, R/πR) is the of π : A A, so 0. → + We want to apply this to the module G(λ) µ for λ X(T ) and µ X(H). We need to note that this module∇ is defined⊗ over R∈, that is, there ∈ is an R-free HR-module M such that the Hk-module Mk is isomorphic to G(λ) µ (on identifying Hk and H). This is clear for G(λ), since dual Weyl∇ modules⊗ are even defined over Z, and for the 1-dimensional∇ H-module µ since HR is a split . The theorem now follows from the claim by Theorem 1.4 for (i) and (ii) and Corollary 1.3 for (iii). We note that the conditions in parts (ii) and (iii) of the theorem are not necessary conditions for the dense orbit to be preserved on reduction modulo p – compare Proposition 3.4 and Lemma 1.5 for examples.

3. Good pairs We now consider how to verify the cohomological condition in Theorem 2.2. In fact, we use an even stronger condition to do with good filtrations which turns out to be easier to verify. Recall that a G-module V has a good filtration if it has an ascending filtration 0 = V0 < V1 < V2 < with V = i 0 Vi and each Vi/Vi 1 ··· + S ≥ − isomorphic to some G(λ) with λ X(T ) . In that case, the dimension ∇ ∈ of HomG(∆G(λ),V ) simply computes the number of factors in any such filtration that are isomorphic to G(λ), by [13, II.4.16(a)]. The dual of a good filtration is a Weyl filtration∇. There is also a cohomological criterion [13, II.4.16(b)]: V has a good 1 + filtration if H (G, V G(λ)) = 0 for all λ X(T ) . In particular, this ⊗ ∇ ∈ 266 cohomological criterion implies that if V1 < V2 are G-modules with good filtrations, then V2/V1 also has a good filtration ([13, II.4.17]). We say that (G, H) is a good pair if H is a closed, connected reductive subgroup of G such that the restriction to H of every G-module V with a good filtration has a good filtration as an H-module. We say that (G, H) is a good integral embedding if it is both a good pair and an integral embedding. By the cohomological criterion for good filtrations, if (G, H) is a good 1 integral embedding, then it satisfies the condition H (H, G(λ) µ) = 0 in Theorem 2.2 (ii), for all λ, µ. ∇ ⊗ We record the important theorem due to Donkin [7] (in almost all cases) and Mathieu [19] (in general):

Theorem 3.1 (i) If V1,V2 are G-modules with good filtrations, then V1 V2 has a good filtration. ⊗ (ii) For any Levi subgroup L of G, the pair (G, L) is a good pair. In particular, the theorem implies that the pair (G, L) for any Levi subgroup of G is a good integral embedding. To apply Theorem 2.2 to more general subgroups than Levi subgroups, we need some further examples of good integral embeddings. To construct these, the following basic facts will be useful: Lemma 3.2 (i) If (G, H) and (H,L) are good pairs, so is (G, L). (ii) (G, H) is a good pair if and only if (G, H0) is a good pair. (iii) If (G1,H1) and (,H2) are good pairs, so is (G1 G2,H1 H2). (iv) For G semisimple and simply-connected, (G, H) is× a good× pair if and only if G(ωi) H has a good filtration for all i I. (v) If V ∇is a finite-dimensional↓ G-module with a∈ Weyl filtration, and v is a non-zero vector of maximal weight λ, then the G-submodule W of V generated by v is isomorphic to ∆G(λ), and V/W has a Weyl filtration. Proof. Part (i) is obvious and (ii) follows from [7, Proposition 3.2.7 (ii)]. For (iii), argue as in [7, Proposition 3.4.6] and for (iv) use [7, Proposition 3.5.4] combined with Theorem 3.1 (i). Finally, (v) is a result of Jantzen; see [7, 11.5.1]. Proposition 3.3 The (simply-connected covers of the) following pairs are good integral embeddings: (i) (G, H) = (SL2n(k), Sp2n(k)) (n > 1); (ii) (G, H) = (Sp2n+2m(k), Sp2n(k)Sp2m(k)) (n, m 1); ≥ (iii) (G, H) = (SLn(k),SOn(k)) (n 1) if p = 2; ≥ 6 (iv) (G, H) = (SOn+m(k),SOn(k)SOm(k)) (n, m 1) if either p = 2 or one of n or m is at most 2; ≥ 6 (v) (G, H) = (SO8(k), Spin7(k)); (vi) (G, H) = (D4,G2) or (B3,G2); 267

(vii) (G, H) = (G2,A2) or (G2,A1A˜1). A proof is given in the appendix. We clarify some of the embeddings: the embedding of Spin7 in SO8 is the simply-connected group B3 embedded via a spin representation. In (vi), G2 is the centralizer of a triality in a simply connected D4, and lies in a subgroup of D4 of type B3. Finally, in the maximal rank subgroup A1A˜1 < G2 in (vii), A1 denotes the long root SL2, and A˜1 denotes the short root one. We conclude the section with a negative result showing that the re- strictions in Proposition 3.3 (iii) and (iv) are necessary. This is interesting because it gives examples of maximal rank subgroups (coming from a closed subsystem of the root system) that are not good, in types B or D. On the other hand, all such subgroups yield good pairs in all characteristics in types A, C or G; I do not know about E and F . Proposition 3.4 Let p = 2. Then the following are not good pairs: (i) (G, H) = (SLn(k),SOn(k)) (n 3); ≥ (ii) (G, H) = (SOn+m(k),SOn(k)SOm(k)) (n, m > 2). Proof. For (i), see the remark on [7, p. 75]. Now consider (G, H) = (SOn+m(k),SOn(k)SOm(k)). If n, m are both odd, let V be the natural G- module. If V H has a good filtration, it contains a dual Weyl module for ↓ H as a submodule, necessarily one of SO (ω1) or SO (ω1) by character ∇ n ∇ m considerations; but this contradicts the fact that the socle of V H is 1- dimensional. It remains to consider the case where one of n, m is even;↓ here we will only sketch the argument. Let G be simply connected of type Bs+t or Ds+t and let H be the maximal rank subgroup BsDt (s 1, t > 1) or DsDt (s, t > 1) respectively. ≥ Let g, h be the corresponding Lie algebras over C and U(g)Z,U(h)Z be their Kostant Z-forms coming from the choice of Chevalley basis made in [12, p. 38]. This choice of basis also fixes a triangular decomposition 0 + U(g)Z = U(g)Z−U(g)ZU(g)Z . We fix the standard basis e 1, . . . , e (s+t), ± ± together with e0 in type B, for the natural g-module V , again as in [12] or [4, Section 2]. The Z-span VZ of this basis is precisely the Z-form U(g)Z−e1 for V , and U(h)Z is a subalgebra of U(g)Z stabilizing the decomposition VZ = e s, . . . , es e (s+1), . . . , e (s+t) . The chosen triangular decomposition h − i ⊕ h ± ± i of U(g)Z determines a corresponding decomposition of U(h)Z. 2 Now, the irreducible g-module V (ω2) is precisely V ; let V (ω2)Z de- 2 V note the Z-form U(g)−.e1 e2 in V . By [12, pp. 38–44], the following Z ∧ V vectors give a basis for V (ω2)Z over Z:

e i e j 1 i

Using this, a direct calculation shows that V (ω2)Z is generated as a U(h)Z−-module by e1 es+1, es+1 es+2, e1 e 1 + es+1 e (s+1), e1 e2 if ∧ ∧ ∧ − ∧ − ∧ H = B1Db, e1 e0 if H = B1Db and e1 e 2 if H = D2Db, but by no proper 6 ∧ ∧ − subset of these vectors. However, the vector 2(e1 e 1 + es+1 e (s+1)) lies ∧ − ∧ − in the U(h)Z−-submodule generated by all these vectors excluding e1 e 1 + ∧ − es+1 e (s+1). Now,∧ − tensoring up with k or C, we have constructed a minimal set of weight vectors generating V (ω2)k as a U(h)k−-module which contains one more vector than a minimal set of weight vectors generating V (ω2)C as a U(h)C−-module. But this implies that V (ω2)k ∼= ∆G(ω2) does not have a Weyl filtration as an H-module, since in that case any such minimal set over k has the same size as over C because of Lemma 3.2 (v).

4. Spherical subgroups Now we discuss one special case, namely the classification of reductive spherical subgroups. As we mentioned in the introduction, all reductive spherical subgroups of G have been classified over C in [2, 16, 20]. In posi- tive characteristic, many examples of spherical subgroups are known thanks to a result of Springer [27, Lemma 4.1] which shows that, providing p = 2, any centralizer K of an involution of G is spherical. Such subgroups K6 , or rather the associated symmetric varieties G/K, have been studied by many authors (see e.g. [10, 17, 23]). In [16], Kr¨amerclassified all pairs (GR,HR) of compact, connected Lie groups, with GR simple and HR < GR, satisfying the analogous represen- tation-theoretic condition to the one in Theorem 1.3. Equivalently by [14], Kr¨amer’sresults classify all pairs (GC,HC) of connected reductive algebraic groups over C with GC simple such that HC is a spherical subgroup of GC. We list all such pairs (GC,HC) from Kr¨amer’sclassification in table 1. We note that the property that HC is spherical in GC is preserved by of GC, so we only list the embeddings for our favourite representative of the root datum of GC. In the left hand side of table 1, we have listed all pairs for which HC is (the connected component of) a centralizer of an involution of GC, and the remaining cases are listed in the right hand side. In the table, we denote a 1-dimensional central torus of H by T1. Also note that the embeddings of Spin7 < SO9 and Spin7SO2 < SO10 are via Spin7 < SO8 < SO9 and Spin7SO2 < SO8SO2 < SO10 respectively. All the embeddings in table 1 can be considered equally well over our field k of arbitrary characteristic (the only ambiguity being SO2n+1(k) < SL2n+1(k) if p = 2, where we mean either of the two classes of such a sub- group). There is at present no complete classification of reductive spherical subgroups in characteristic p. We give here two partial results. 269

Table 1. Kr¨amer’sspherical subgroups

GC HC GC HC

SLn+m SLnSLmT1 n, m 1 SLn+m SLnSLm n = m ≥ 6 SL2n Sp2n n 2 SL2n+1 Sp2nT1 n 1 ≥ ≥ SLn SOn n 2 SL2n+1 Sp2n n 1 ≥ ≥ Sp2n+2m Sp2nSp2m n, m 1 Sp2n+2 Sp2nT1 n 1 ≥ ≥ Sp2n GLn n 1 SO2n+1 GLn n 2 ≥ ≥ SOn+m SOnSOm n, m 1 SO4n+2 SL2n+1 n 1 ≥ ≥ SO2n GLn n 2 SO7 G2 ≥ SO8 G2 SO8 Sp2 Sp4 ⊗ SO8 Spin7 SO9 Spin7 G2 A1A˜1 SO10 Spin7SO2 T1D5 G2 A2 T1E6 E6 D5 B4 F4 A1C3 E6 F4 E6 C4 E6 A1A5 E7 A7 E7 A1D6 D8 E8 A1E7

First, we have the classification of spherical Levi subgroups in arbitrary characteristic, which is immediate by Kr¨amer’sresult and Theorem 2.2: r Theorem 4.1 Let L be a Levi subgroup of G. Let G0 = Gi as a Qi=1 commuting product of simple factors and set Li := L Gi. Then, L is ∩ spherical in G if and only if, for each i, either Gi = Li or (Gi,Li0 ) is one of (An,AmAn m 1), (Bn,Bn 1), (Bn,An 1), (Cn,Cn 1), (Cn,An 1), − − − − − − (Dn,Dn 1), (Dn,An 1), (E6,D5) or (E7,E6). − − This result also gives a classification of all parabolic subgroups P of G for which a Borel subgroup of a Levi factor of P has a dense orbit (equivalently, finitely many orbits) on the unipotent radical of P , acting by conjugation. This follows because of the following observation (see [2, Proposition 1.1] for a more general result): Lemma 4.2 Let P be a parabolic subgroup of G with P = LV , where V = Ru(P ), and let BL be a Borel subgroup of L. There is a dense L-orbit on G/B if and only if there is a dense BL-orbit on V . 270

Proof. We may assume that the Borel subgroup B of G is contained + + in P and that BL = B L. Let P = LV be the opposite parabolic to P . Now, L has a dense∩ orbit on G/B if and only if L B has a dense orbit on P +B, since P +B is dense in G. The map P + ×V P +B given by multiplication is an isomorphism of varieties, so this× is→ equivalent to L B having a dense orbit on P + V , where the action of L is by left × + × multiplication on P , and the action of an element ub B (b BL, u V ) + 1 1 1 ∈ ∈ ∈ on (p, v) P V is ub.(p, v) = (pb− , bvb− u− ). Since the orbits of L B + ∈ × + × on P V under this action are of the form LpBL V for p P , this is × + × ∈ if and only if L BL has a dense orbit on P , i.e. BL has a dense orbit on P +/L. Now the× quotient map V + P +/L is an isomorphism of varieties, → + so this is equivalent to BL having a dense orbit on V , and hence V . In fact, for all of the pairs listed in Theorem 4.1 except for (Bn,An 1) − and (Cn,Cn 1), the unipotent radical V of P is abelian. This abelian unipo- tent radical− case has been studied by a number of authors: for instance, in [24], the number of L-orbits on V + is shown to equal the number of P,P - double cosets in G. Also, Vavilov [28, 4] has given a direct proof that there + § are finitely many BL-orbits on V here. The proof of the next theorem has only recently been completed, thanks to work of Lawther [18]. Lawther’s results show that there are finitely many double cosets for all the cases in the bottom half of table 1, even in characteristic 2. Of course, in characteristic different from 2, these cases are centralizers of involutions, so finiteness follows from Springer’s result. Lawther’s approach in characteristic 2 involves bounding the number of double cosets for the corresponding finite groups of Lie type, using Deligne- Lusztig . Theorem 4.3 The entries in table 1 are spherical in all characteristics.

Proof. The cases (SLn,SOn) and (SOn+m,SOnSOm) are verified in Lemma 1.5, and (SO8, Sp2 Sp4) follows from (SO8,SO3SO5) on applying triality. The cases in the bottom⊗ half of table 1 follow from [27, Lemma 4.1](p = 2) or [18] (p = 2). The remaining entries in table 1 are immediate from Theorem6 2.2, Lemma 3.2, Theorem 3.1 (ii) and the list of good pairs in Proposition 3.3. We conclude with a conjecture motivated by the article; there is some hope for a case-free proof of this (for part (i) it only remains to show that (E6,F4) is a good pair, given Proposition 3.3). Conjecture 4.4 Let H be a closed reductive subgroup of G. If either (i) H is the centralizer of a graph of G; or (ii) H is the centralizer of an involution of G, then (G, H) is a good pair. 271

A. Proof of Proposition 3.3 In this appendix, we give the proof of Proposition 3.3. We will verify each case in the proposition in turn, using Lemma 3.2 (iv). Part (i) is immediate from the following result of Donkin:

Lemma A.1 ([9, Appendix A]) Let G = Sp2n(k), with natural module V . Then for 1 r n, r V has a filtration ≤ ≤ V r r r 0 = ( V )0 < < ( V )m = V V ··· V V such that r r G(ω2i) r even, ( V )i/( V )i 1 =  ∇ V V − ∼ G(ω2i+1) r odd, ∇ for all 0 < i m, where m = [ r ]. ≤ 2 Consider the pair (G, H) = (Sp2n+2m(k), Sp2n(k)Sp2m(k)). We have just shown that (SL2n(k), Sp2n(k)) is a good pair, so Lemma 3.2 (iii) implies that (SL2n(k)SL2m(k), Sp2n(k)Sp2m(k)) is a good pair. Also, by Theorem 3.1 (ii) and Lemma 3.2 (ii), (SL2n+2m(k),SL2n(k)SL2m(k)) is a good pair. Combining, it follows that (SL2n+2m(k), Sp2n(k)Sp2m(k)) is a good pair. Now the conclusion follows from the next lemma:

Lemma A.2 Let G = SL2n(k), H = Sp2n(k). Suppose L < H < G is a connected reductive algebraic group. Then (H,L) is a good pair if and only if (G, L) is a good pair. Proof. The forward implication is immediate by Lemma 3.2 (i), so con- sider the converse. By Lemma 3.2 (iv), we need to check that H (ωi) has a good filtration as an L-module for each 1 i n. For ∇i = 1, ≤ ≤ V = H (ω1) = G(ω1) H , so has a good filtration as an L-module as ∇ ∼ ∇ ↓ i (G, L) is a good pair. In general, G(ωi) = V has a good filtration as an L-module by assumption, and∇ contains anVH-submodule isomorphic i 2 to − V by Lemma A.1. This submodule has a good filtration as an L- moduleV by assumption, hence so does the quotient. But this quotient is precisely H (ωi), completing the proof. ∇ Next, we look at Proposition 3.3 (iii), (iv) and (v). If p = 2 and G 6 is the simply-connected cover of SOn(k), we note that the fundamental modules G(ωi) are all irreducible, by for instance [12, Satz I.13]. So, ex- cluding the∇ spin modules, these modules coincide with the exterior power i V , where V is the natural G-module. Now Proposition 3.3 (iii) and V(iv) for p = 2 are obvious using Lemma 3.2 (iv). For p = 2, we need 6 to show (SOn+m(k),SOn(k)SOm(k)) is a good pair if m 2. If m = 2, this is just a Levi subgroup, so the result follows from Theorem≤ 3.1. If m = 1, this is proved in [4, Theorem D] (unfortunately though the proof 272 in [4] for D` < B` is not quite complete, though the result is correct). The case (SO8(k), Spin7(k)) follows from (SO8(k),SO7(k)) on applying triality (working in the simply-connected cover of SO8(k) where triality is defined). For the remaining two cases, we will use the fact that if G is a simple, simply-connected group, then Lie(G) is a Weyl module. First, for Proposi- tion 3.3 (vi), let G be the simply connected group of type D4 and H = G2. We can view H as L L , where L is a simply-connected group of type ∩ 0 B3 coming from the natural embedding SO7(k) < SO8(k), and L0 is the image of L under a triality. Consider the natural G-module ∆G(ω1) of di- mension 8. Since ∆G(ω1) L contains the 7-dimensional module ∆L(ω1) as a submodule, it has a Weyl↓ filtration on restriction to H. Now apply- ing triality yields that ∆G(ω3) and ∆G(ω4) also have Weyl filtrations as H-modules. So, it remains to show that ∆G(ω2) ∼= Lie(G) has a Weyl fil- tration on restriction to H. A direct check using Chevalley bases shows that Lie(G) = Lie(L)+Lie(L0), hence that Lie(L) Lie(L0) = Lie(L L0) = Lie(H) by dimension. Now we claim that 0 < Lie(∩ H) < Lie(L) <∩Lie(G) is a Weyl filtration as an H-module. Certainly, Lie(H) ∼= ∆H (ω2). Since (G, L) is a good pair by (iv), Lemma 3.2 (v) implies that Lie(G)/Lie(L) is a Weyl module, necessarily ∆L(ω1), which has a Weyl filtration as an H-module. The same argument shows that Lie(G)/Lie(L0) has a Weyl fil- tration as an H-module; but Lie(G)/Lie(L0) = Lie(L) + Lie(L0)/Lie(L0) ∼= Lie(L)/Lie(L) Lie(L0) = Lie(L)/Lie(H). These arguments also show that (L, H) is a good∩ pair. Now let G = G2. To prove (vii), there are two G-modules to check for Weyl filtrations: the 7-dimensional module ∆G(ω2) and the 14-dimensional . For H = A2, ∆G(ω2) is semisimple as an H-module (factors of dimension 1, 3, 3) and the module Lie(G) contains Lie(H) = ∆H (ω1 + ω2) as a submodule, and the quotient is semisimple (factors of dimension 3, 3); so both have Weyl filtrations here. For H = A1A˜1, we note that ∆G(ω2) H ↓ has the same character as ∆A (ω1) ∆ ˜ (ω1) ∆ ˜ (2ω1), so it is a direct 1 ⊗ A1 ⊕ A1 sum of irreducible Weyl modules as an A1-module, and has a Weyl filtration as an A˜1-module by Theorem 3.1; this is enough to show (using Lemma 3.2 (v)) that it has a Weyl filtration as an A1A˜1-module. Also, by considering weights we see that the Lie algebra splits as a of Lie(H) together with a module with the same character as ∆A (ω1) ∆ ˜ (3ω1), so this 1 ⊗ A1 summand is again a direct sum of irreducible Weyl modules as an A1- module and has a Weyl filtration as an A˜1-module by Theorem 3.1. Hence it has a Weyl filtration as an A1A˜1-module as desired. This completes the proof of Proposition 3.3. 273

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